Conventionally, communication receivers use two types of MPSK modulated signal detection: coherent detection and differential detection. In coherent detection, a carrier phase reference is detected at the receiver, against which subsequent symbol phases are compared to estimate the actual information phase. Differential detection processes the difference between the received phases of two consecutive symbols to determine the actual phase. The reference phase is the phase of the first of the two consecutive symbols, against which the difference is taken. Although differential detection eliminates the need for carrier phase reference processing in the receiver, it requires a higher signal-to-noise ratio at a given symbol error rate.
Differential detection in an Additive White Gaussian Noise (AWGN) channel is preferred over coherent detection when simplicity of implementation and robustness take precedence over receiver sensitivity performance. Differential detection is also preferred when it is difficult to generate a coherent demodulation reference signal. For differential detection of multiple-phase shift keying (MPSK) modulation, the input phase information is differentially encoded at the transmitter, then demodulation is implemented by comparing the received phase between consecutive symbol intervals. Therefore, for proper operation, the received carrier reference phase should be constant over at least two symbol intervals.
Multiple-symbol differential detection (MSDD) uses more than two consecutive symbols and can provide better error rate performance than conventional differential detection (DD) using only two consecutive symbols. As in the case of DD, MSDD requires that the received carrier reference phase be constant over the consecutive symbol intervals used in the process.
Detailed discussions of MSDD and Multiple Symbol Detection (MSD) are found in, “Multiple-Symbol Differential Detection of MPSK” (Divsalar et al., IEEE TRANSACTIONS ON COMMUNICATIONS, Vol. 38, No. 3, Mar. 1990) and “Multiple-Symbol Detection for Orthogonal Modulation in CDMA System” (Li et al., IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, Vol. 50, No. 1, January 2001).
Conventional MPSK MSDD is explained in conjunction with FIGS. 1 and 2 below. FIG. 1 shows an AWGN communication channel 101 with an MPSK signal sequence r that comprises N consecutive symbols r1 . . . rN received by receiver 110. Symbol rk represents the kth component of the N length sequence r, where 1≦k≦N. The value for rk is a vector represented by Equation (1):
                              r          k                =                                                                              2                  ⁢                                      E                    S                                                                    T                  S                                                      ⁢                          ⅇ                                                jϕ                  k                                +                                  jθ                  k                                                              +                      n            k                                              Eq        .                                  ⁢                  (          1          )                    having symbol energy Es, symbol interval Ts and transmitted phase φk where j=√{square root over (−1)}. Value nk is a sample taken from a stationary complex white Gaussian noise process with zero mean. Value θk is an arbitrary random channel phase shift introduced by the channel and is assumed to be uniformly distributed in the interval (−π, π). Although channel phase shift θk is unknown, differential detection conventionally operates assuming θk is constant across the interval of observed symbols r1 to rN. For differential MPSK (DMPSK), phase information is differentially encoded at the transmitter, and transmitted phase φk is represented by:φk=φk-1+Δφk  Eq. (2)where Δφk is the transmitted information phase differential corresponding to the kth transmission interval that takes on one of M uniformly distributed values within the set Ω={2 πm/M, m=0, 1, . . . , M−1}around the unit circle, as in a Gray mapping scheme. For example, for QPSK, M=4 and Δφk=0, π/2, π, or 3π/2 for each k from 1 to N.
It is assumed for simplicity that arbitrary phase value θk is constant (θk=θ) over the N-length of the observed sequence.
At the receiver, optimum detection using multiple-symbol differential detection (MSDD) is achieved by selecting an estimated sequence of phase differentials {d{circumflex over (φ)}1, d{circumflex over (φ)}2, . . . , d{circumflex over (φ)}N−1} which maximizes the following decision statistic:
                    η        =                              max                                          d                ⁢                                                                  ⁢                                                      ϕ                    ^                                    1                                            ,                              d                ⁢                                                                  ⁢                                                      ϕ                    ^                                    2                                            ,                                                d                  ⁢                                                                          ⁢                                                            ϕ                      ^                                                              N                      -                      1                                                                      ∈                Ω                                              ⁢                                                                                    r                  1                                +                                                      ∑                                          m                      =                      2                                        N                                    ⁢                                                            r                      m                                        ⁢                                          ⅇ                                                                        -                          j                                                ⁢                                                                                                  ⁢                        d                        ⁢                                                                                                  ⁢                                                                              ϕ                            ^                                                                                m                            -                            1                                                                                                                                                                                  2                                              Eq        .                                  ⁢                  (          3          )                    
By Equation (3), the received signal is observed over N symbol time intervals while simultaneously selecting the optimum estimated phase sequence {d{circumflex over (φ)}1, d{circumflex over (φ)}2, . . . , d{circumflex over (φ)}N−1}. The maximized vector sum of the N-length signal sequence rk, provides the maximum-likelihood detection, where estimated phase differential d{circumflex over (φ)}m is the difference between estimated phase {circumflex over (φ)}m+1 and the estimate of the first phase {circumflex over (φ)}1.d{circumflex over (φ)}m={circumflex over (φ)}m+1−{circumflex over (φ)}1.  Eq. (4)The estimate of transmitted information phase sequence {Δ{circumflex over (φ)}1, Δ{circumflex over (φ)}2, . . . , Δ{circumflex over (φ)}N−1} is obtained from the estimated phase sequence {d{circumflex over (φ)}1, d{circumflex over (φ)}2, . . . , d{circumflex over (φ)}N−1} using Equation (5).
                              d          ⁢                                          ⁢                                    ϕ              ^                        m                          =                              ∑                          k              =              1                        m                    ⁢                      Δ            ⁢                                                  ⁢                                          ϕ                ^                            k                                                          Eq        .                                  ⁢                  (          5          )                    Value Δ{circumflex over (φ)}k is an estimate of transmitted phase differential Δφk. Since d{circumflex over (φ)}k (1≦k≦N−1) takes on one of M uniformly distributed Ω values {2 πm/M, m=0, 1, . . . , M−1}, the conventional MSDD detection searches all possible phase differential sequences and there are MN−1 such phases. The error rate performance improves by increasing the observed sequence length N, which preferably is selected to be N=4 or N=5. As an example, for 16PSK modulation with N=5, the number of phase differential sequences to search is 164=65536. As evident by this considerably large number of sequences, simplicity in the search sequence is sacrificed in order to achieve a desirable error rate performance.
FIG. 2 shows the process flow diagram for algorithm 200, which performs conventional MSDD. It begins with step 201 where N consecutive symbols rk for k=1 to N is observed. Next, the possible sets of phase differential sequences {d{circumflex over (φ)}1, d{circumflex over (φ)}2, . . . , d{circumflex over (φ)}N−1} where each d{circumflex over (φ)}k, for k=1 to N−1, is one from the set of M uniformly distributed phase values in the set Ω={2 πm/M, m=0, 1, . . . , M−1}. There are MN−1 possible sets. FIG. 5 shows an example of an array of such sets, where N=4 and M=4, which illustrates the 44-1=64 possible sets of phase differential sequences. In step 203, each possible phase sequence is attempted in the expression
                                    r          1                +                              ∑                          m              =              2                        N                    ⁢                                    r              m                        ⁢                          ⅇ                                                -                  j                                ⁢                                                                  ⁢                d                ⁢                                                                  ⁢                                                      ϕ                    ^                                                        m                    -                    1                                                                                                  2    ,giving a total of MN−1 values. Next, in step 204, the maximum value is found for step 203, which indicates the best estimate phase differential sequence. Finally, in step 205, the final information phase sequence {Δ{circumflex over (φ)}1, Δ{circumflex over (φ)}2, . . . , Δ{circumflex over (φ)}N−1} is estimated from {d{circumflex over (φ)}1, d{circumflex over (φ)}2, . . . , d{circumflex over (φ)}N−1} using Equation (5) and the information bits are obtained from Gray de-mapping between phase and bits.
Although MSDD provides much better error performance than conventional DD (symbol-by-symbol), MSDD complexity is significantly greater. Therefore, it is desirable to provide an improved method and system for MSDD with less complexity.